Constant-selection evolutionary dynamics on weighted networks

TL;DR

This work investigates how edge weights in undirected networks affect constant-selection evolutionary dynamics under birth-death and death-Birth updating. By computing fixation probabilities via a Markov-chain framework on weighted graphs, it demonstrates that random edge weights largely suppress selection compared to the Moran process, and that symmetric weighted graphs can either amplify or suppress depending on weight configurations. The results show a robust tendency for weighted networks to be less amplifying than their unweighted counterparts, with many six-node networks acting as suppressors and empirical networks largely following the suppression pattern. The findings highlight edge weights as a practical lever to engineer fixation dynamics and motivate perturbation and optimization approaches for weighted networks in evolutionary contexts.

Abstract

The population structure often impacts evolutionary dynamics. In constant-selection evolutionary dynamics between two types, amplifiers of selection are networks that promote the fitter mutant to take over the entire population, and suppressors of selection do the opposite. It has been shown that most undirected and unweighted networks are amplifiers of selection under a common updating rule and initial condition. Here, we extensively investigate how edge weights influence selection on undirected networks. We show that random edge weights make small networks less amplifying than the corresponding unweighted networks in a majority of cases and also make them suppressors of selection (i.e., less amplifying than the complete graph, or equivalently, the Moran process) in many cases. Qualitatively, the same result holds true for larger empirical networks. These results suggest that amplifiers of selection are not as common for weighted networks as for unweighted counterparts.

Figures (6)

• Figure 1: Visualization of amplifiers and suppressors of selection relative to the Moran process. The different lines represent the fixation probability as a function of the fitness of the mutant, $r$. We set the fitness of the resident to $1$. The vertical dotted line marks $r=1$, at which the mutant and resident types have the same strength and the fixation probability is equal to $1/N$ for all the three curves.
• Figure 2: Fixation probability of weighted networks on six nodes, for the Bd and dB updating. The horizontal axis represents the index of the 112 non-isomorphic unweighted networks on six nodes. For each unweighted network, we assigned random edge weights to generate a weighted network 20 times. The vertical axis represents the difference from the Moran process in terms of the fixation probability. (a) Bd rule, $r=0.9$. (b) Bd rule, $r=1.3$. (c) dB rule, $r=0.9$. (d) dB rule, $r=1.3$. A magenta circle represents a weighted network. A black square represents the corresponding unweighted network.
• Figure 3: Fixation probability of weighted complete graphs with a single edge with a different edge weight. (a) An example network with $N=8$ nodes. The single edge with weight $w \neq 1$ is shown by the thick line. (b)--(d) Fixation probability relative to that of the Moran process for the weighted complete graphs of the type shown in (a). (b) $N=4$. (c) $N=10$. (d) $N=150$. In (b)--(d), the dotted lines represent $r=1$.
• Figure 4: Fixation probability of weighted complete graphs with two larger groups of nodes. (a) An example network with $N_1 = 4$ and $N_2 = 3$. The edges with weight $1$, $w_1$, and $w_2$ are shown in solid, dashed, and dotted lines, respectively. (b) and (c): Fixation probability relative to that of the Moran process for the weighted complete graphs of the type shown in (a); the dotted lines represent $r=1$. We set $N_1=N_2=5$ in (b) and $N_1=N_2=25$ in (c). (d) and (e): Fixation probability of the same type of weighted complete graphs with $N_1=N_2=50$ and varying $w_1$ and $w_2$. (d) $r=0.9$. (e) $r=1.3$.
• Figure 5: Fixation probability of weighted star graphs. (a) A weighted star graph with $N_1 = 2$ and $N_2 = 4$. The edges with weight $w$ are shown in thick lines. (b) Fixation probability relative to that of the Moran process for the weighted star graphs with $N=4$ and $N_1=1$ as $w$ varies. (c) Same for $N=12$ and $N_1=3$. (d) Same for $N=40$ and $N_1 = 10$. (e) Fixation probability relative to that of the Moran process for various weighted star graphs on 20 nodes. In (b)--(e), the dotted lines represent $r=1$. (f) and (g): Fixation probability of weighted star graphs with $N=40$ and various $N_1$ and $w$ values. (f) $r=0.9$. (g) $r=1.3$. Note that $1 \le N_1 \le N-2 = 38$. In (f) and (g), the black horizontal lines and the green arrows pointing to them in the color bar show the values for the unweighted star graph.